a. Let X1,..., Xn be i.i.d. random variables from N(u1,) and Y1,.., Xm be i.i.d. random variables from N(u2, 0) with o unknown. Suppose we want to test the hypothesis Ho : µ1 = µ2. Under Ho the test statistic follows a central t distribution with n+m - 2 degrees of freedom. Why? Explain how the power of the test is computed in this problem. You will need to determine the noncentrality parameter and include it in your answer. Note: Assume H. : µ1 > µ2. b. Consider the simple regression model y; = Bo + B1x; + €;, i = 1,...,n with e; ~ N(0, 0). The es are independent. Develop the likelihood ratio test for testing Ho : B1 = 0 against Ha : B1 # 0. c. Suppose X o2, and that the covariance matrix of X can be expressed as E = o²V, where o? is known and V is nxn symmetric matrix of known constants. Find a confidence interval for µ using 1-a confidence level. (X1, X2, ..., X,n)' follows a multivariate normal distribution with E(X;) = µ, var(X;) = d. Let Q ~ beta(;,(n – k – 2)), where n and k are integers. Show that (n - k – 2) follows an F distribution. What are the degrees of freedom?